Arguments

Logical flag. If TRUE then the Akaike Information
Criterion is used to choose the order of the autoregressive
model. If FALSE, the model of order order.max is
fitted.

order.max

Maximum order (or order) of model to fit. Defaults
to 10*log10(N) where N is the number
of observations except for method="mle" where it is the
minimum of this quantity and 12.

method

Character string giving the method used to fit the
model. Must be one of the strings in the default argument
(the first few characters are sufficient). Defaults to
"yule-walker".

na.action

function to be called to handle missing values.

demean

should a mean be estimated during fitting?

series

names for the series. Defaults to
deparse(substitute(x)).

var.method

the method to estimate the innovations variance
(see Details).

...

additional arguments for specific methods.

object

a fit from ar.

newdata

data to which to apply the prediction.

n.ahead

number of steps ahead at which to predict.

se.fit

logical: return estimated standard errors of the
prediction error?

Details

For definiteness, note that the AR coefficients have the sign in

(x[t] - m) = a[1]*(x[t-1] - m) + ... + a[p]*(x[t-p] - m) + e[t]

ar is just a wrapper for the functions ar.yw,
ar.burg, ar.ols and ar.mle.

Order selection is done by AIC if aic is true. This is
problematic, as of the methods here only ar.mle performs
true maximum likelihood estimation. The AIC is computed as if the variance
estimate were the MLE, omitting the determinant term from the
likelihood. Note that this is not the same as the Gaussian likelihood
evaluated at the estimated parameter values. In ar.yw the
variance matrix of the innovations is computed from the fitted
coefficients and the autocovariance of x.

ar.burg allows two methods to estimate the innovations
variance and hence AIC. Method 1 is to use the update given by
the Levinson-Durbin recursion (Brockwell and Davis, 1991, (8.2.6)
on page 242), and follows S-PLUS. Method 2 is the mean of the sum
of squares of the forward and backward prediction errors
(as in Brockwell and Davis, 1996, page 145). Percival and Walden
(1998) discuss both. In the multivariate case the estimated
coefficients will depend (slightly) on the variance estimation method.

Remember that ar includes by default a constant in the model, by
removing the overall mean of x before fitting the AR model,
or (ar.mle) estimating a constant to subtract.

Value

For ar and its methods a list of class "ar" with
the following elements:

order

The order of the fitted model. This is chosen by
minimizing the AIC if aic=TRUE, otherwise it is order.max.

ar

Estimated autoregression coefficients for the fitted model.

var.pred

The prediction variance: an estimate of the portion of the
variance of the time series that is not explained by the
autoregressive model.

x.mean

The estimated mean of the series used in fitting and for
use in prediction.

x.intercept

(ar.ols only.) The intercept in the model for
x - x.mean.

aic

The value of the aic argument.

n.used

The number of observations in the time series.

order.max

The value of the order.max argument.

partialacf

The estimate of the partial autocorrelation function
up to lag order.max.

resid

residuals from the fitted model, conditioning on the
first order observations. The first order residuals
are set to NA. If x is a time series, so is resid.

method

The value of the method argument.

series

The name(s) of the time series.

frequency

The frequency of the time series.

call

The matched call.

asy.var.coef

(univariate case, order > 0.)
The asymptotic-theory variance matrix of the coefficient estimates.

For predict.ar, a time series of predictions, or if
se.fit = TRUE, a list with components pred, the
predictions, and se, the estimated standard errors. Both
components are time series.